To determine the inverse of the function [tex]\( f(x) = 2x - 5 \)[/tex], we follow a methodical approach. Here is a step-by-step solution:
1. Start with the function:
[tex]\[ f(x) = 2x - 5 \][/tex]
2. Replace [tex]\( f(x) \)[/tex] with [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 5 \][/tex]
3. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = 2y - 5 \][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[ x + 5 = 2y \][/tex]
[tex]\[ y = \frac{x + 5}{2} \][/tex]
5. Rewrite [tex]\( y \)[/tex] as [tex]\( f^{-1}(x) \)[/tex]:
[tex]\[ f^{-1}(x) = \frac{x + 5}{2} \][/tex]
Now, we look at the provided options to see which matches our derived inverse function:
- [tex]\[ f^{-1}(x) = \frac{x+2}{5} \][/tex]
- [tex]\[ f^{-1}(x) = \frac{x+5}{2} \][/tex]
- [tex]\[ f^{-1}(x) = \frac{x-2}{5} \][/tex]
- [tex]\[ f^{-1}(x) = \frac{x-5}{2} \][/tex]
The correct option that matches our derived function is:
[tex]\[ f^{-1}(x) = \frac{x+5}{2} \][/tex]
Thus, the inverse of the function [tex]\( f(x) = 2x - 5 \)[/tex] is
[tex]\[ f^{-1}(x) = \frac{x+5}{2}. \][/tex]
Therefore, the correct answer is [tex]\(\boxed{f^{-1}(x) = \frac{x+5}{2}}\)[/tex].
